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Mirrors > Home > MPE Home > Th. List > tnggrpr | Structured version Visualization version GIF version |
Description: If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
tngngp3.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
Ref | Expression |
---|---|
tnggrpr | ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngp3.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
2 | eqid 2724 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | tngbas 24475 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
4 | eqidd 2725 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | eqid 2724 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | tngplusg 24477 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
7 | 6 | eqcomd 2730 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝑇) = (+g‘𝐺)) |
8 | 7 | oveqdr 7430 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝑇)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
9 | 3, 4, 8 | grppropd 18873 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp ↔ 𝐺 ∈ Grp)) |
10 | 9 | biimpd 228 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp → 𝐺 ∈ Grp)) |
11 | ngpgrp 24432 | . 2 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
12 | 10, 11 | impel 505 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 Grpcgrp 18855 NrmGrpcngp 24410 toNrmGrp ctng 24411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-tset 17217 df-ds 17220 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-ngp 24416 df-tng 24417 |
This theorem is referenced by: tngngp3 24497 |
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